Stabilization of vibrating beam with a tip mass controlled by combined feedback forces

摘要

A flexible structure consisting of a Euler-Bernoulli beam with a tip mass is considered. To stabilize this system we use a boundary control laws: -u(xxx)(l, t) + mu(tt)(l, t) = - alphau(t)(l, t) + betau(xxxt)(l, t) and u(xx)(l, t) = - gammau(xt)(l, t). A sensitivity asymptotic analysis of the system's eigenvalues and eigenfunctions is set up. We prove that all of the generalized eigenfunctions of (2.9) form a Riesz basis of H By a new method, we prove that the operator A generates a C-0 contraction semigroup T(t), t greater than or equal to 0. Furthermore T(t), t greater than or equal to 0, is uniformly exponentially stable and the optimal exponential decay rate can be obtained from the spectrum of the system. (C) 2001 Academic Press. [References: 12]